An Introduction to Scilab for EE 210 Circuits Tony Richardson

Transcription

1 An Introduction to Scilab for EE 210 Circuits Tony Richardson Introduction This is a brief introduction to Scilab. It is recommended that you try the examples as you read through this introduction. What the program shows is in Courier font. What user types is shown in bold Courier font Basic Operations Define a scalar (a single number) and a matrix: -->a = 2; --> [ 1 2; 3 4] // Spaces between cols, semicolons between rows! 1. 2.!! 3. 4.! a is a scalar, while A is a 2x2 matrix whose first row contains the elements 1 and 2. Matrices are entered in row order with spaces (or commas) used to separate columns and semicolons (or a carriage return) between rows. By default the result of the definition is displayed unless the line of input ends with a semicolon (as when a is defined above). Comments begin with a double slash (//). We can access a particular element by specifying indices inside parenthesis (row #, column #): -->A(1,1) = 6 // Change a single element in the matrix! 6. 2.!! 3. 4.! -->B(3,3) = 5 // Define a new matrix by defining a single element B =! !! !! ! The functions zeros(m,n), ones(m,n), eye(m,n) and rand(m,n) can be used to define m x n matrices of all zeros, all ones, the identity matrix, and a matrix of random numbers. The special symbols %i, %pi, and %e represent 1, , and : --> [2+2*%i cos(2*%pi); log(%e) %e^(%i*%pi)] 09/12/07 1 of 6

2 ! i 1.!! E-16i! The ':' (colon) operator can be used to form a vector whose elements increase (or decrease) incrementally: -->x = [1:6] x =! ! -->y = [1:2:9] y =! ! You can also use vectors as indices to select a submatrix: --> rand(4,4) // Define a new random matrix! !! !! !! ! -->A ([1 2],1) // Select first two elements from the first column! !! ! -->A([3:$],[2:$]) // Rows 3-4 of columns 2-4 ($ is last row or col)! !! ! Matrix Operations Scilab's native data type is the matrix, so the +, -, and * operators perform matrix addition, subtraction, and multiplication. --> [ --> > >] // Define a matrix separating rows by newlines! !! ! 09/12/07 2 of 6

3 -->B = A' B = // B is the transpose of A (rows and columns swapped)! 1. 4.!! 2. 5.!! 3. 6.! -->A*B // Multiply matrix A times B! !! ! -->A + A // Add A to itself! !! ! -->A.* A // Element-by-element multiplication! !! ! Notice that element-by-element multiplication uses.* instead of *. Getting Help Scilab has an extensive on line help system. Click on the Help in the menu bar or type help at the command line. You can get help on a specific topic by typing help colon for example, to get help on the use of the colon operator. The Scilab User's Manual (intro.pdf) and Reference Manual (manual.pdf) provide complete information. Solving Systems of Linear Equations We can represent the following system of linear equations: i 1 2i 2 i 3 =1 2i 1 i 2 i 3 =3 i 1 2i 2 3i 3 =7 as the matrix equation R i = v, where i is an unknown array of three currents. We define matrices R and v in Scilab as follows: -->R = [1 2-1; 2-1 1; ] R =! !! !! ! 09/12/07 3 of 6

4 -->v = [1 3 7]' v =! 3.!! 7.! Here the transpose operator (') was used to conveniently transpose the [1 3 7] row vector into a column vector. (I find this to be more convenient than typing semicolons at the end of each row of a column vector.) Scilab provides the det() function for finding the determinant of a matrix, so we can also easily use Cramer's rule to find the elements of the i array. -->i1 = det([v R(:,2:3)])/det(R) i1 = 1. -->i2 = det([r(:,1) v R(:,3)])/det(R) i2 = 1. -->i3 = det([r(:,1:2) v])/det(r) i3 = 2. In computing i1, the notation [v R(:,2:3)] is used to create a new matrix whose first column is v and whose second and third columns are the second and third columns of R. In computing i2 we need the determinant of the matrix formed by replacing the middle column of R with v and in computing i3 we need the determinant of the matrix formed by replacing the last column of R with v. While using Cramer's rule works, solving the matrix equation R i = v for i gives i = R -1 v. In Scilab we find i directly as follows: -->i = inv(r)*v i =! 2.! Note that these are the same values obtained previously. We also can verify that this is indeed the solution by comparing R i and v. -->R*i 09/12/07 4 of 6

5 ! 3.!! 7.! Here is a complete example from circuit analysis. Nodal analysis is to be used to determine the node voltages in the circuit shown in Figure Ω 3 ma v 1 v 2 v Ω 1500 Ω 8 ma 3000 Ω 25 ma 600 Ω Figure 1 We can write the nodal equations directly as: v 1 v v 1 v = v v 2 v v 2 v = v v 3 v v 3 v = Rewriting these equations in standard form yields: v v v 3 = v v v 3 = v v v 3 = Instead of using a calculator to determine the value of expressions such as, 1/ /750, I find it easier to just let Scilab compute the values for me. With practice you can enter the relevant Scilab matrices directly from the circuit diagram. The above system of equations can be written in matrix form as G v = i. Entering the conductance matrix G and the current array i into Scilab gives: -->G = [ --> 1/1000+1/750-1/1000-1/750 09/12/07 5 of 6

6 --> -1/1000 1/1000+1/3000+1/1500-1/ > -1/750-1/1500 1/750+1/1500+1/600 -->] G =! !! !! ! -->i = [-11e-3 3e-3 25e-3]' i =! !! 0.003!! 0.025! Solving the matrix equation G v = i for v gives v = G -1 i or in Scilab: -->v = inv(g)*i v =! 3.!! 6.!! 9.! So the node voltages are 3 V, 6 V, and 9 V respectively. 09/12/07 6 of 6